Uniform local weak o-minimality and $*$-local weak o-minimality

TL;DR

This paper introduces new notions of uniform local weak o-minimality and $*$-local weak o-minimality, establishing local monotonicity theorems and dimension formulas in these structures, extending previous results in o-minimal theory.

Contribution

It develops the concepts of uniform local weak o-minimality and $*$-local weak o-minimality, proving local monotonicity theorems and dimension results under new conditions.

Findings

Established local monotonicity theorems for these structures.

Proved dimension formulas hold in $*$-locally weakly o-minimal structures.

Extended properties from locally o-minimal to these new structures.

Abstract

We propose the notions of uniform local weak o-minimality and $*$-local weak o-minimality. Local monotonicity theorems hold in definably complete locally o-minimal structures and uniformly locally o-minimal structures of the second kind. In this paper, we demonstrate new local monotonicity theorems for uniformly locally weakly o-minimal structures of the second kind and for locally o-minimal structures under the assumption called the univariate $*$-continuity property. We also prove that several formulas for dimension of definable sets which hold in definably complete locally o-minimal structures also hold in $*$-locally weakly o-minimal structures possessing the univariate $*$-continuity property.